Magnetic resonance imaging (MRI) is a medical imaging modality that can create images of the inside of a human body without using x-rays or other ionizing radiation. MRI uses a powerful magnet to create a strong, uniform, static magnetic field (i.e., the “main magnetic field”). When a human body, or part of a human body, is placed in the main magnetic field, the nuclear spins that are associated with the hydrogen nuclei in tissue water become polarized. This means that the magnetic moments that are associated with these spins become preferentially aligned along the direction of the main magnetic field, resulting in a small net tissue magnetization along that axis (the “z axis,” by convention). A MRI system also comprises components called gradient coils that produce smaller amplitude, spatially varying magnetic fields when current is applied to them. Typically, gradient coils are designed to produce a magnetic field component that is aligned along the z axis and that varies linearly in amplitude with position along one of the x, y or z axes. The effect of a gradient coil is to create a small ramp on the magnetic field strength, and concomitantly on the resonance frequency of the nuclear spins, along a single axis. Three gradient coils with orthogonal axes are used to “spatially encode” the MR signal by creating a signature resonance frequency at each location in the body. Radio frequency (RF) coils are used to create pulses of RF energy at or near the resonance frequency of the hydrogen nuclei. These coils are used to add energy to the nuclear spin system in a controlled fashion. As the nuclear spins then relax back to their rest energy state, they give up energy in the form of an RF signal. This signal is detected by the MRI system and is transformed into an image using a computer and known reconstruction algorithms.
During a transmit sequence, an MR system will transmit RF pulses of given frequencies having particular amplitude profiles while various magnetic field gradients are being applied. For example, in a spin-echo sequence, an MR system would transmit an excitation pulse at a particular frequency, amplitude and transmit power for a particular time, in order to induce a net transverse magnetization in nuclei of a scan subject. Subsequent pulses transmitted by the system may have the same or similar frequency, but might have different gain, amplitude, and duration attributes to cause a different change in magnetization (or “flip angle”) in order to cause spin echoes. Thus, in general, the particular shapes of the RF pulses in a transmit sequence are varied to manipulate the net magnetization in nuclei of the scan subject.
MR systems may transmit RF pulses using a single channel (or coil) or using multiple channels (known as parallel transmission). In parallel transmission, each coil exhibits a spatially different sensitivity pattern and is driven by an independently controlled RF waveform generator with a separate power amplifier. The different spatial sensitivity patterns of the coils provide additional ability of spatial selectivity in addition to the inherent spatial selectivity from RF pulses. The additional spatial selectivity has been used to reduce pulse length or to reduce error in the resulting magnetization. Two primary applications of parallel transmission are reduced field of view (FOV) excitation and B1 inhomogeneity correction.
Many parallel transmission pulse design methods are based on the small-tip-angle (STA) approximation. In the STA approximation, it is assumed that the initial magnetization vector points in +z direction (positive longitudinal direction) and that the flip angle is small (e.g., <30° or at most 90° depending on the error tolerance and the k-space trajectory used). Based on these assumptions, the Bloch equation is reduced to the following linear equation (which ignores relaxation):
                                          p            ⁡                          (              r              )                                =                      ⅈ            ⁢                                                  ⁢            γ            ⁢                                          ∑                                  l                  =                  1                                L                            ⁢                                                          ⁢                                                                    s                    l                                    ⁡                                      (                    r                    )                                                  ⁢                                                      ∫                    0                    T                                    ⁢                                                                                    b                        l                                            ⁡                                              (                        t                        )                                                              ⁢                                          ⅇ                                              ⅈ                        ⁢                                                                                                  ⁢                                                                              k                            ⁡                                                          (                              t                              )                                                                                ·                          r                                                                                      ⁢                                                                                  ⁢                                          ⅆ                      t                                                                                                          ,                            Eqn        .                                  ⁢        1            
where p(r) is the desired normalized transverse magnetization profile (i.e., p(r)=Mxy,desired(r)/M0) at spatial location r=(x,y,z), γ is gyromagnetic ratio, T is pulse duration, k(t) denotes the chosen excitation k-space trajectory, sl(r) represents the transmit sensitivity of the lth coil, and bl(t) denotes the RF pulse waveform to be designed for the lth coil, l=1, 2, . . . , L. By discretizing the time axis as 0, Δt, 2Δt, . . . , (N−1) Δt (where N is the number of time points and Δt is the step size) and space axes as r1, r2, . . . , rM (where M is the number of voxels), and denoting pm=p(rm), m=1, 2, . . . , M, Eqn. 1 can be converted into a matrix form:
                                                        [                                                                                          p                      1                                                                                                                                  p                      2                                                                                                            ⋮                                                                                                              p                      M                                                                                  ]                                      ︸              P                                =                                    ∑                              l                =                1                            L                        ⁢                                                  ⁢                          ⅈ              ⁢              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                                        )                                                                                                  ·                                                                  r                                  1                                                                                                                                                                                                                …                                                                                                                                                        s                              l                                                        ⁡                                                          (                                                              r                                1                                                            )                                                                                ⁢            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 r                                2                                                            )                                                                                ⁢                                                      ⅇ                                                          ⅈ                              ⁢                                                                                                                          ⁢                                                                                                k                                  ⁡                                                                      (                                    0                                    )                                                                                                  ·                                                                  r                                  2                                                                              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                                                        N                                          -                                          1                                                                                )                                                                            ⁢                                      Δ                                      ⁢                                                                                                                                                          ⁢                                      t                                                                        )                                                                                                  ]                                                            ·                                                              r                                2                                                                                                                                                                                                                        ⋮                                                                    ⋮                                                                    ⋱                                                                    ⋮                                                                                                                                                                                                s                              l                                                        ⁡                                                          (                                                              r                                M                                                            )                                                                                ⁢                                                      ⅇ                                                          ⅈ                              ⁢                                                                                                                          ⁢                                                                                                k                                  ⁡                                                                      (                                    0                                    )                                                                                                  ·                                                                  r                                  M                                                                                                                                                                                                                                                                                                    s                              l                                                        ⁡                                                          (                                                              r                                M                                                            )                                                                                ⁢                                                      ⅇ                                                          ⅈ                              ⁢                                                                                                                          ⁢                                                                                                k                                  ⁡                                                                      (                                                                          Δ                                      ⁢                                                                                                                                                          ⁢                                      t                                                                        )                                                                                                  ·                                                                  r                                  M                                                                                                                                                                                                                …                                                                                                                                                        s                              l                                                        ⁡                                                          (                                                              r                                M                                                            )                                                                                ⁢                                                      ⅇ                                                                                                                            ⅈ                                  ⁢                                                                                                                                          ⁢                                                                      k                                    [                                                                                                                  (                                                                                  N                                          -                                          1                                                                                )                                                                            ⁢                                      Δ                                      ⁢                                                                                                                                                          ⁢                                      t                                                                        )                                                                                                  ]                                                            ·                                                              r                                M                                                                                                                                                                                          ]                                                  ︸                                      S                    l                                                              ⁢                                                [                                                                                                                                          b                            l                                                    ⁡                                                      (                            0                            )                                                                                                                                                                                                                    b                            l                                                    ⁡                                                      (                                                          Δ                              ⁢                                                                                                                          ⁢                              t                                                        )                                                                                                                                                              ⋮                                                                                                                                                                  b                            l                                                    ⁡                                                      (                                                                                          (                                                                  N                                  -                                  1                                                                )                                                            ⁢                              Δ                              ⁢                                                                                                                          ⁢                              t                                                        )                                                                                                                                ]                                                  ︸                                      b                    l                                                                                      ⁢                                  ⁢                  Or          ,                                          ⁢                      in            ⁢                                                  ⁢            short                    ,                                    Eqn        .                                  ⁢        2                                          p          =          Sb                ,                            Eqn        .                                  ⁢        3            where p is a length-M vector (a column vector), S=[S1, S2, . . . , SL] is an M×LN matrix, and b=[(b1)T, (b2)T, . . . , (bL)T]T (unitalicized T denotes matrix transpose) is a length-LN vector. To solve Eqn. 3, and RF power-regularized least square formulation is used resulting in:Choose b to minimize ∥Sb−p∥w2+λ∥b∥22,  Eqn. 4The weighted l2 norm ∥Sb−p∥w2=(Sb−p)HW(Sb−p) (H denotes the Hermitian transpose) in Eqn. 4 for a given spatial weighting matrix W=diag{w1, w2, . . . , wM} (wm≧0, ∀m) represents the overall error of the final magnetization profile. The l2 norm ∥b∥22=bHb represents the RF power and λ is a positive scalar (referred to as a regularization parameter) that balances the error and RF power terms. The quadratic objective function in Eqn. 4 can be solved by a standard conjugate gradient (CG) method or by explicit matrix inversion:b*=(SHWS+λI)−1SHWp.  Eqn. 5.
The desired magnetization p in Eqn. 4 contains a desired magnitude profile and a constant phase profile (e.g., zero phase). This is overly constrained because a parallel transmission array does not necessarily favor excitation with flat phase. Therefore, it is generally advantageous to relax the constant phase constraint in Eqn. 4 in parallel transmission RF pulse design when the phase profile of the final magnetization is not of interest, as is the case in most of the MRI applications. In one known phase relaxed parallel transmission pulse design, the phase of one of the eigenmodes of the transmit sensitivities is chosen that has the minimum magnitude error as the initial predetermined target phase profile for p and then the phase profile is iteratively update with the phase of Sb in the CG iterations.
Conventional STA parallel transmission designs can have the following limitations: 1) an un-optimized magnitude profile of the transverse magnetization as a result of enforcing an overly restrictive flat phase profile and 2) only small tip angle pulses can be accurately designed. Various STA based parallel transmission RF pulse designs have been developed to address the first limitation by using a phase relaxed design using predetermined constant or non-constant phase profiles. Methods to design large tip angle or arbitrary tip angle parallel transmission RF pulses have been developed that utilize optimal control design.
It would be desirable to provide a system and method capable of producing accurate parallel transmission (or multi-channel) RF pulse waveforms with an optimal phase profile. It would also be desirable to have embodiments of such system and method which could account for large tip angles, small tip angles or arbitrary tip angles.